Finite Group Databasebeta

← #135461

Subgroups — #135461

Order 512.

431 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20525C211yesmaximal, characteristic
2#11418C211yesmaximal, characteristic
3C2×#1314C211yesmaximal, characteristic
4#7202C212yesmaximal, complemented
5#7200C212yesmaximal, complemented
6C2×#407C411yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8C2×C4×C22⋊C4C411yescharacteristic
9#2112C2212yes—
10C2×C22⋊C16C2212yes—
11#1753—21no—
12#1753—21no—
13#1752C411yescharacteristic
14#1752C2211yescharacteristic
15C8×C22⋊C4C411yescharacteristic
16C8×C22⋊C4C2211yescharacteristic
17#1749—21no—
18C4×C22⋊C8—21no—
19#1314—21no—
20#1314—21no—
21#1314—21no—
22#1314—21no—
23C24×C4C2×C411yescharacteristic
24C22×M16—21no—
25C23×C8C2×C411yescharacteristic
26C23×C8C2311yescharacteristic
27C22×C22⋊C4—21no—
28C22×C42—21no—
29C2×M32—22no—
30C22×C16—22no—
31#407—22no—
32#407—22no—
33C2×C22⋊C8—21no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8C2×C411yescharacteristic
37C2×C22⋊C8C2×C411yescharacteristic
38C2×C22⋊C8D411yescharacteristic
39C2×C22⋊C8D411yescharacteristic
40C2×C22⋊C8D411yescharacteristic
41C2×C22⋊C8D411yescharacteristic
42C2×#97—21no—
43C2×#97—21no—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C4×C22⋊C4—21no—
47C4×C22⋊C4—21no—
48C4×C22⋊C4C811yescharacteristic
49C4×C22⋊C4C811yescharacteristic
50C4×C22⋊C4C2×C411yescharacteristic
51C4×C22⋊C4C2×C411yescharacteristic
52C2×#95C811yescharacteristic
53C2×#95C811yescharacteristic
54C22⋊C16—22no—
55C22⋊C16—22no—
56C22⋊C16—22no—
57C22⋊C16—22no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C25M1611yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—81no—
70C23×C4—81no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C2×C811yescharacteristic
74C23×C4C2×C811yescharacteristic
75C23×C4M1611yescharacteristic
76C23×C4C22×C411yescharacteristic
77C2×M16—24no—
78C2×M16—81no—
79C2×M16—81no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—81no—
85C22×C8—81no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C22×C8#4112yes—
91C22×C8#4112yes—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—44no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—81no—
102C2×C42—81no—
103M32—82no—
104C2×C16—42no—
105C2×C16—42no—
106C2×C16—82no—
107C22⋊C8—24no—
108C22⋊C8—21no—
109C22⋊C8—21no—
110C22⋊C8—22no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113C22⋊C8—22no—
114C22⋊C8M1612yes—
115C22⋊C8M1612yes—
116#97—81no—
117#97—81no—
118C4×C8—81no—
119C4×C8—81no—
120#95—21no—
121#95—21no—
122#95C2×C811yescharacteristic
123#95C2×C811yescharacteristic
124#95C2×C811yescharacteristic
125#95C2×C811yescharacteristic
126C24—21no—
127C24—21no—
128C24—22no—
129C24—41no—
130C24—42no—
131C24—81no—
132C24C22⋊C811yescharacteristic
133C24C22⋊C811yescharacteristic
134C24C2×M1611yescharacteristic
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4—81no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4C22⋊C811yescharacteristic
171C22×C4C22⋊C811yescharacteristic
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4#11711yescharacteristic
174C22×C4#11711yescharacteristic
175C22×C4C22×C811yescharacteristic
176C22×C4C2×M1611yescharacteristic
177C22×C4C2×M1611yescharacteristic
178M16—84no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8#12312yes—
196C2×C8#12312yes—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C42—41no—
202C42—41no—
203C42—41no—
204C42—41no—
205C42—81no—
206C42—81no—
207C16—82no—
208C16—82no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—22no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—41no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23—42no—
226C23—81no—
227C23—81no—
228C23—84no—
229C23—84no—
230C23#35012yes—
231C23C2×C22⋊C811yescharacteristic
232C23#43211yescharacteristic
233C23#43211yescharacteristic
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4#38111yescharacteristic
274C2×C4#40711yescharacteristic
275C2×C4#43111yescharacteristic
276C2×C4#43111yescharacteristic
277C8—22no—
278C8—22no—
279C8—42no—
280C8—42no—
281C8—41no—
282C8—41no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—84no—
301C22—81no—
302C22—81no—
303C22—84no—
304C22#181711yescharacteristic
305C22C2×#35011yescharacteristic
306C22#211711yescharacteristic
307C22#216512yes—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4#182811yescharacteristic
318C4#182911yescharacteristic
319C2—21no—
320C2—21no—
321C2—22no—
322C2—41no—
323C2—42no—
324C2—81no—
325C2#873411yescharacteristic
326C2#1284811yescharacteristic
327C2#1284911yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1817
prime radicalC25M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7202C2semidirect
5. #7200C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#205252
#114182
C2×#13142
#72022
#72002